Louisville (20-5, 8-4 Big East) trailed 71-64 with just under 6 minutes left, but Kuric (17 points) highlighted an 11-0 run with a game-tying, long-range bullet as time ticked down.
The Mountaineers' Truck Bryant then turned it over, and Russ Smith (16 points) put the Cardinals ahead to stay, 73-71, on a layup with 2:17 left.
Kuric sealed the win when he intercepted a Gary Browne pass with 5 seconds left and was fouled, making both shots for the final margin.
Louisville remained without a loss since a Jan. 16 setback at Marquette and now goes on to visit Syracuse.
West Virginia (16-10, 6-7) was led by Kevin Jones win 22 points and 11 rebounds.
The Mountaineers fell in their third straight home contest.